Current evidence identifies two inter-related systems for representing and manipulating quantities in humans: 1) a symbolically-mediated exact numerical system used in precise mathematical operations; 2) a nonverbal approximate sense of numerosity that is evolutionarily primitive and which is present early in human ontogeny. Neurobiological and behavioral data strongly suggest that the symbolically-mediated exact numerical system taps into the approximate numerosity sense. Sensitivity to numerosity, a fuzzy sense of the number of objects or events, predicts numerical and mathematical performance throughout development. Moreover, brain imaging and lesion studies in humans implicate parietal cortex in both exact numerical processing and the approximate numerosity sense. Single neurons in the primate ventral intraparietal area (VIP), as well as in prefrontal cortex (dlPFC), respond selectively to a specific number of elements in a visual array. By contrast, neurons in the lateral intraparietal area (LIP) respond in a monotonic fashion to the number of elements in a visual array located within the neuronal receptive field. These observations suggest the hypothesis that LIP neurons integrate visual information to form a representation of accumulated numerosity, which is read out by neurons in VIP and dlPFC to signal a specific numerical value. This hypothesis is consistent with several computational models of numerical representation, in which numerosity units representing a specific cardinal value, such as 3 or 5, receive input from summation units encoding the quantity of elements within their receptive fields. Despite the attractiveness of this hypothesis, several important questions remain. First, neurons representing cardinal numerosity were described in monkeys trained to make explicit same/different judgments, while neurons representing accumulated numerical magnitude were described in monkeys that were not trained to make any explicit numerosity judgments. Thus, the neuronal coding scheme used to represent number may reflect training or task demands, rather than the intrinsic numerical coding properties of neurons in number-sensitive areas. Since math performance can be improved by training on simple numerosity and spatial manipulation tasks, understanding the effects of explicit training on numerosity encoding by neurons in parietal cortex is particularly important. Second, the assumption that summation units in LIP provide inputs to numerosity units in VIP (or dlPFC), and that these connections are functionally relevant for numerosity discrimination, remains to be tested. The goal of the proposed project is to address these questions using behavioral, neurophysiological, pharmacological, and computational techniques. Understanding the neural mechanisms that relate visuospatial information-processing to representations of numerosity may suggest important improvements in early childhood mathematical education and remediation that will benefit all children, particularly those suffering from impaired quantitative abilities.